Lab 1

1/4/2022

About me

A little bit about me. As you heard on Tuesday, I am a fourth year graduate student in the Psych & Brain Sciences department. (click) My research focuses on exercise and how it influences cognition. I specifically look at how the neural mechanisms of working memory and attention are modulated as a result of an acute bout of aerobic exercise.

A fun fact about me is that I love hiking! This is a photo from a while back when I hiked Cucomonga peak down in the San Bernadino mountains. Unfortunately, I haven’t been able to hike as much during grad school but I am trying to change that with the hopefully few years I have left.

I also am very passionate about statistics, especially the Bayesian approach, machine learning, and AI. One of my hobbies is trying to use reinforcement learning methods to train video game AI to perform optimally. I say hobby because I’m not that good at it.

Purpose of labs

Facilitate learning theoretical concepts covered in lecture
Answer questions about homework
Review for midterms and the final
Implement statistical analyses from lecture in RStudio

Lab 1 Outline

Cover preliminary statistics and probability concepts
Intro to R and setup

Random Variable

A variable that takes on specific values with specific probabilities.
Measurable functions that map outcomes of a stochastic process to a measurable space.
Typically denoted by a capital letter (e.g., X, Y, Z).
Denote the outcome of a coin flip as Y
- Y = 1 if Heads, else Y = 0
- p(Y=1) = 0.5

Probability distribution

Probability mass function (PMF)
- Assigns probabilities to individual values of a discrete random variable

Probablity distribution

Probability density function (PDF)
- Similar to a PMF, but instead specifies the probability that a continuous variable takes on a range of values.

Normal Distribution Notation

X ~ N(\(\mu\),\(\sigma^2\))

Expected value and Variance

The expected value of a random variable Y is denoted as E(Y).
- Probability weighted average of all possible values.

y <- c(70,80,85,90,100)
p.y <- c(0.18,0.34,0.35,0.11,0.02)

E.y <- sum(y*p.y)
E.y

## [1] 81.45

Variance
- A measure of how disperse all possible values of a random variable are from the expected value (i.e. population or sample mean)

Multivariate Distributions

PDF \(\rightarrow\) Joint Probability Density

For the random variables X and Y, the joint pdf characterizes the probability that each X and Y takes on a set of values.

Multivariate Distributions

Variance \(\rightarrow\) Covariance

Measure of how much two random variables vary together.
Formally, is the expected value of each random variables deviation from its respective mean.

cov(X,Y) = E[(X - E[X])(Y - E[Y])]
Typically represented as a matrix.

Marginal Distribution

Probability distribution of an outcome for one random variable in the presence of all other outcomes for another random variable

	x₁	x₂	x₃	x₄	p_Y(y_i)
y₁	0.125	0.0625	0.03125	0.03125	0.25
y₂	0.09375	0.1875	0.09375	0.09375	0.46875
y₃	0.28125	0	0	0	0.28125
p_X(x_i)	0.5	0.25	0.125	0.125	1

Pearson correlation & Statistical independence

Pearson correlation coefficient \(\rho\) measures the linear relationship between two random variables
Two random variables are statistically independent if the realization of one does not affect the outcome of the other.

Likelihood

Throughout this course we are going to use statistical models (e.g., regression, ANOVA) to describe patterns of variability in random variables.
These models have parameters (e.g., mean of a sampling distribution)
A likelihood function is the joint probability of observed data as a function of parameters in a statistical model.

R

Programming Tips

Google is your best friend!
More than a single way to skin a cat.
Learn more than one language.
Have fun!